Why can't I translate parts of my mesh with the 3d widget?
There are three main transformations available: Translate, Rotate, and Scale.

Keyboard shortcuts:


Grab (G) (aka translate/move)
Rotate (R)
Scale (S)


From the wiki:


  the normal Transform commands (G for Grab, R for Rotation, S for Scale), can be used to manipulate objects along any axis.


Widgets:

You can also use the the transform manipulator widgets to do this by pressing on the arrows with LMB.

You can enable or disable the transform manipulator widget by pressing the  icon in 3D view > Header:



Or by pressing CtrlSpace.

You can change the transofm manipulator type by selecting one of the buttons again in the Header of the 3D view:


Location () 
Rotation ()
Scale ()


It sounds like you want Location manipulators, but you have Scale type selected.

Try setting it like this by pressing one of the icons:



Also note that you can select multiple widgets at once by holding Shift while selecting them.